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### Coordinate systems

• 1. Lectures on Classical Mechanics ( J. R. Taylor) By G. HASNAIN TARIQ Department of Physics, KFUEIT, Rahim Yar Khan
• 3. Mechanics: The branch of Physics concerned with motion and forces producing motion. Galileo (1564-1642) Newton (1642-1727) Lagrange (1736 -1813) Hamilton (1805- 1865) Up to the beginning of the twentieth century, it seemed that classical mechanics was the only kind of mechanics, correctly describing all possible kinds of motion. 1905 to 1925, it became clear that classical mechanics neither correctly describe the motion of objects moving at speeds close to the speed of light, nor that of the microscopic particles inside atoms and molecules. The result was the development of two completely new forms of mechanics: relativistic mechanics to describe very high-speed motions and quantum mechanics to describe the motion of microscopic particles. The formulation of Newton is based on his three laws of motion. Newton's three laws of motion are formulated in terms of four crucial underlying concepts: space, time, mass, and force.
• 4. Space: This is the concept by which we define the location of objects and events. It is the concept of point, position, direction and displacement. In a three-dimensional space each point P can be labeled by a position vector “r” which specifies the distance and direction of P from a chosen origin “O”as in Figure. There are many different ways to identify a vector. We can use to express this by three unit vectors, 𝒙, 𝒚, 𝒛, or (i, j, k) pointing along the three axes and to write; r = x 𝒙 + 𝑦 𝒚 + 𝑧 𝒛 ……………………… (1) r = (x, y, z) ……………………………… (2) r is the vector whose components are x, y, z. For most vectors, we indicate the components by subscripts x, y, z. Thus the velocity vector v has components vx , vy , vzand the acceleration a has components ax , ay , az. We can label the three components x, y, z of r as r1 , r2 , r3 , and the three unit vectors 𝒙, 𝒚, 𝒛 as e1 , e2 , e3 . That is, we define r1 = x, r2 = y, r3 = z and e1 = 𝒙, e2 = 𝒚, e3 = 𝒛 . The symbol e is commonly used for unit vectors, since e stands for the German "eins“ or "one“. Eq. 1 becomes; r = r1e1+r2e2 + r3e3 = 𝑖=1 3 (𝑟𝑖 𝒆𝑖) ……………………….. (3) Fig. 1: The point P is identified by its position vector r
• 5. Vector Operations: • If r and s are vectors with components r = (r1 , r2 , r3) and s = (s1 , s2 , s3),then their sum (or resultant) r + s is; found by adding corresponding components, so that r + s = (r1 + s1, r2 + s2 , r3 + s3) .................................................... (4) • When two forces Faand Fb act on an object, the effect is the same as a single force, the resultant force, which is just the vector sum F = Fa+ Fb • If c is a scalar (that is, an ordinary number) and r is a vector, the product cr is given by ; cr = (cr1 , cr2 , cr3)……… (5) ⸫cr is a vector in the same direction as r with magnitude equal to c times the magnitude of r. There are two important kinds of product that can be formed from any pair of vectors. • First, the scalar product (or dot product) of two vectors r and s is given by either of the equivalent formulas r • s = rs cos θ …………………………… (6)r • s = r1 s1 + r2 s2 + r3 s3 = 𝑛=1 3 ( rn s 𝑛 ) ……………………. (7) where r and s denote the magnitudes of the vectors r and s, and θis the angle between them. The magnitude (or length) of any vector r is denoted by 𝒓 or r and, by Pythagoras’s theorem is, 𝑟 = 𝑟1 2 + 𝑟2 2 + 𝑟3 2 = 𝒓. 𝒓 = 𝒓2 = 𝒓 ……………………………………… (8)
• 6. • The second kind of product of two vectors r and s is the vector product (or cross product), which is defined as the vector p = r x s with components, px = rysz – rzsy , py = rzsx - rx sz , pz = rxsy - ry sx ..................................... (9) r x s is a vector perpendicular to both r and s, with direction given by the familiar right-hand rule and magnitude rs sin θ. Reference Frames: Almost every problem in classical mechanics involves a choice (explicit or implicit) of a reference frame, that is, a choice of spatial (relating to space) origin and axes to label positions as in Fig. 1 and a choice of temporal (relating to time) origin to measure times. In certain special frames, called inertial frames, the basic laws (Newton's first law, the law of inertia) hold true in their standard, simple form. In an inertial frame of reference an isolated body, would move with uniform velocity . If a second frame is accelerating or rotating relative to an inertial frame, then this second frame is noninertial, and the basic laws — in particular, Newton’s laws — do not hold in their standard form in this second frame. Consider three reference frames, S, Sʹ, Sʺ. S is an inertial frame, where an object is placed. How Sʹ or Sʺ will be inertial? How Sʹ or Sʺ will be noninertial?!
• 7. Fundamental Assumptions One of the fundamental assumptions of physics is that space and time are continuous; therefore • An event occurred at a specific point in space and a specific instant of time. • It is also meaningful to say that there are universal standards of length and time. This means that observers in different places and at different times can make meaningful comparisons of their measurements. In “classical” physics, it is assumed further that: • There is a universal time scale, meaning that observers who have synchronized their clocks will always agree about the time of any event. • The geometry of space is Euclidian, the three dimensional model that we ordinarily use. • There is no limit in principle to the accuracy with which we can measure positions and velocities.
• 8. Newton's First and Second Laws; Inertial Frames Newton's First Law (the Law of Inertia) In the absence of forces, a particle moves with constant velocity v. In the absence of forces, a stationary particle remains stationary and a moving particle continues to move with unchanging speed in the same direction (uniform velocity). This is, of course, exactly the same as saying that the velocity is always constant. Again, v is constant if and only if the acceleration a is zero, so an even more compact statement is this: In the absence of forces a particle has zero acceleration. Newton's Second Law For any particle of mass m, the net force F on the particle is always equal to the mass m times the particle's acceleration: 𝑭 𝒏𝒆𝒕 = 𝑚𝒂 In this equation 𝑭 𝒏𝒆𝒕 denotes the vector sum of all the forces on the particle and 𝒂is the particle's acceleration, 𝒂 = 𝒅𝒗 𝒅𝒕 = 𝒗 = 𝒅 𝟐 𝒓 𝒅𝒕 𝟐 = 𝒓 The second law can be restated in terms of the particle's momentum, defined as; P = mv ⟹ 𝑷= m 𝒗 = ma Thus second law can be restated, 𝑭 𝒏𝒆𝒕= 𝑷
• 9. 1st Law 2nd Law (Uniform velocity)
• 10. Newton’s Third Law and Conservation of Momentum 1 F21 2 1 2 F12 F21 = - F12 1 2 Fig. 2:Two objects If an object 1 exerts a force F21 on object 2, then object 2 always exerts a reaction force F12 on object 1 given by, F12 = - F21 Forces acting along the line joining 1 and 2. Forces with this extra property are called central forces. (They act along the line of centers.) The third law is intimately related to the law of conservation of momentum. In figure 2 two objects exert forces on each other and may also be subject to additional "external" forces 𝐅1 𝑒𝑥𝑡 and 𝐅2 𝑒𝑥𝑡 , then net forces acting on both objects are; Net force on object 1 = F1= F12 + 𝐅1 𝑒𝑥𝑡 and Net force on object 2=F2 = F21 + 𝐅2 𝑒𝑥𝑡 Now rates of change of the particles' momenta are; 𝑷1 = F1= F12 + 𝐅1 𝑒𝑥𝑡 𝑷2=F2 = F21 + 𝐅2 𝑒𝑥𝑡 The total momentum of two objects; P = P1+ P2 Then, rate of change of the total momentum is; 𝑷 = 𝑷1 + 𝑷2
• 11. As two internal forces, F21 and F12 cancel out because of Newton's third law, and we are left with external forces, so we have; 𝑷 = 𝐅1 𝑒𝑥𝑡 + 𝐅2 𝑒𝑥𝑡 = 𝐅 𝑒𝑥𝑡 We have result that; If 𝐅 𝑒𝑥𝑡 = 0, then 𝑷 = 0 and P = constant In the absence of external forces, the total momentum of our two-particle system is constant. Which is the principle of conservation of momentum. To evaluate total rate of change of the total momentum; For multi particles system; Consider a system of N particles, The mass of a particle α is mα and its momentum is pα . Each of the other (N — 1) particles can exert a force Fαβ , the force on particle αby particle β, as shown in Fig.3. In addition there may be a net external force 𝐅α 𝑒𝑥𝑡 on particle α. Thus the net force on particle α is; (net force on particleα) = Fα= 𝛽≠𝛼 Fαβ + 𝐅α 𝑒𝑥𝑡 ⸫α = 1 …………………….. N According to Newton’s 2nd Law rate of change of momentum pα of particle αis; 𝑷α= Fα= 𝛽≠𝛼 𝐅αβ + 𝐅α 𝑒𝑥𝑡 Total momentum of our N-particle system is, P = α=1 𝑁 pα Fig.3 β = 1………… (N-1), α ≠ β
• 12. Now rates of change of the α particle’s momentum is; 𝑷 = α 𝑷α ⸫ α = 1 …………………….. N Substituting value of 𝐏α, we have, 𝐏 = α ( β≠α 𝐅αβ + 𝐅α ext ) = α β≠α 𝐅αβ + α 𝐅α ext The double sum here contains N (N - 1) terms in all. Each term 𝐅αβ in this sum can be paired with a second term 𝐅βα, (that is, F12 paired with F21, and so on), so that α β≠α 𝐅αβ = α β>α(𝐅αβ+ 𝐅βα ) each term is the sum of two forces, (𝐅αβ + 𝐅βα ), and, by the third law, each such sum is zero. Therefore, 𝐏 = α 𝐅α ext If 𝐅α ext = 0 , then 𝐏 = 0 and P = constant Hence proved the law of conservation of momentum that, if the net external force 𝐅 𝑒𝑥𝑡 on an N-particle system is zero, the system’s total momentum P is constant.
• 13. Coordinate system In geometry, a coordinate system is a system that uses one or more numbers, or coordinates, to uniquely determine the position of the points or other geometric elements such as Euclidean space. Common coordinate systems are; 1. Number line The simplest example of a coordinate system is the identification of points on a line with real numbers using the number line. In this system, an arbitrary point O (the origin) is chosen on a given line. The coordinate of a point P is defined as the signed distance from O to P, where the signed distance is the distance taken as positive or negative depending on which side of the line P lies. Each point is given a unique coordinate and each real number is the coordinate of a unique point. 2. Cartesian coordinate system 3. Polar coordinate system 4. Cylindrical coordinate system 5. Spherical coordinate system P
• 14. Cartesian Coordinate Systems A Cartesian coordinate system is a coordinate system that specifies each point uniquely in a plane by a set of numerical coordinates. These systems may be; • One-dimensional space – that is, for a straight line (number line)—involves choosing a point O of the line (the origin), a unit of length, and an orientation for the line. • Two dimensions space: (also called a rectangular coordinate system or an orthogonal coordinate system) is defined by an ordered pair of perpendicular lines (axes). Signed distances to the point from two fixed perpendicular directed lines, measured in the same unit of length. Each reference line is called a coordinate axis or just axis (plural axes) of the system, and the point where they meet is its origin, at ordered pair (0, 0). The coordinates can also be defined as the positions of the perpendicular projections of the point onto the two axes, expressed as signed distances from the origin. • Three-dimensional space: Consists of an ordered triplet of lines (the axes) that go through a common point (the origin), and are pair-wise perpendicular; an orientation for each axis; and a single unit of length for all three axes. Each pair of axes defines a coordinate plane. The coordinates are usually written as three numbers surrounded by parentheses and separated by commas, as in (3,4,1). Thus, the origin has coordinates (0,0,0). No standard names of coordinates in the three axes (x-abscissa, y-ordinate and z-applicate are sometimes used).
• 15. P
• 16. Polar coordinate system (2-D) In two dimensions, the cartesian coordinates, (x,y) specify the location of a point P in the plane (coordinates refer to distances along the two coordinate axes). Another two-dimensional coordinate system is polar coordinates. Instead of using the signed distances along the two coordinate axes, polar coordinates specifies the location of a point P in the plane by its distance r from the origin and the angle θ made between the line segment from the origin to P and the positive x-axis. The polar coordinates (r,θ) or (r, φ) of a point P are shown in the figure. Given the rectangular coordinates x and y, we can calculate the polar coordinates r and θ, or vice versa, using the following relations. x = r cos θ y = r sin θ 𝑟 = 𝑥2 + 𝑦2 θ = arctan(y/x)
• 17. Cylindrical Coordinates This is a three-dimensional coordinate system, we take an axis (usually called the z-axis) and a perpendicular plane, on which we choose a ray (the initial ray) originating at the intersection of the plane and the axis (the origin). The coordinates of a point P ( ρ, φ, z) or ( r, θ,z) (radius, azimuth, elevation) are the polar coordinates of the projection of P on the plane and the coordinate z of the projection of P on the axis. Cylindrical coordinates ( r, θ,z)
• 18. Spherical coordinate system is another coordinate system for three dimensional space where the position of a point is specified by three numbers: (i) The radial distance/radius or radial coordinate of that point from a fixed origin, (ii) Its polar angle/colatitude, zenith angle, normal angle, or inclination angle (θ) measured from a fixed zenith direction, and (iii) The azimuth angle (φ) of its orthogonal projection on a reference plane that passes through the origin and is orthogonal to the zenith, measured from a fixed reference direction on that plane. The use of symbols and the order of the coordinates differs between sources. In one system frequently encountered in physics (ρ/r, θ, φ) gives the radial distance, polar angle, and azimuthal angle, whereas in another system used in many mathematics books (ρ/r, φ,θ) gives the radial distance, azimuthal angle, and polar angle. In both systems ρ is often used instead of r. The spherical coordinates of a point P are then defined as follows: The radius or radial distance (ρ/r ) is from the origin O to P. The inclination (or polar angle, θ) is the angle between the zenith direction and the line segment OP. The azimuth (or azimuthal angle, φ) is the angle measured from the azimuth reference direction to the orthogonal projection of the line segment OP on the reference plane.
• 19. The spherical coordinates of a point in the convention (radius r, inclination θ, azimuth φ) can be obtained from its Cartesian coordinates (x, y, z) by the formulae; 𝑟 = 𝑥2 + 𝑦2 + 𝑧2 𝜃 = 𝑎𝑟𝑐𝑐𝑜𝑠 𝑧 𝑥2+𝑦2+𝑧2 = arccos 𝑧 𝑟 𝜑 = 𝑎𝑟𝑐𝑡𝑎𝑛 𝑦 𝑥 Conversely, the Cartesian coordinates may be retrieved from the spherical coordinates (r, θ, φ); 𝑥 = 𝑟𝑠𝑖𝑛𝜃𝑐𝑜𝑠𝜑 𝑦 = 𝑟𝑠𝑖𝑛𝜃𝑠𝑖𝑛𝜑 𝑧 = 𝑟𝑐𝑜𝑠𝜃 Conversion of Spherical & Cartesian Coordinates
• 20. Cylindrical coordinates (radius ρ, azimuth φ, elevation z) may be converted into Spherical coordinates (radius r, inclination θ, azimuth φ), by the formulas; 𝑟 = ρ2 + 𝑧2 𝜃 = 𝑎𝑟𝑐𝑡𝑎𝑛 ρ 𝑧 = 𝑎𝑟𝑐𝑐𝑜𝑠 𝑧 ρ2+𝑧2 = arccos 𝑧 𝑟 𝜑 = 𝜑 Conversion of Cylindrical & Spherical Coordinates Conversely, the Cylindrical coordinates may be retrieved from the Spherical coordinates; ρ = 𝑟𝑠𝑖𝑛𝜃 φ = 𝜑 𝑧 = 𝑟𝑐𝑜𝑠𝜃
• 21. Newton's Second Law in Cartesian Coordinates The three Cartesian components of r are just the appropriate derivatives of the three coordinates x, y, z of r, and the second law (A) become; ………….. (4) Newton’s second law is, F= ma = m 𝒓 ………………………. (A) Which is a second-order, differential equation for the position vector r as a function of the time t. In Cartesian (or rectangular) coordinate system, with unit vectors 𝒙, 𝒚, 𝒛, in terms of which the net force F can then be written as; ………….. (1) and the position vector r as, ….……….. (2) Position vector r in its Cartesian components is easy to differentiate because the unit vectors 𝒙, 𝒚, 𝒛are constant. So we can get, ………..…… (3) F= Fx 𝒙 + 𝐹𝑦 𝒚 + 𝐹𝑧 𝒛 r = x 𝒙 + 𝑦 𝒚 + 𝑧 𝒛 𝒓 = 𝑥 𝒙 + 𝑦 𝒚 + 𝑧 𝒛 Fx 𝒙 + 𝐹𝑦 𝒚 + 𝐹𝑧 𝒛 = m 𝑥 𝒙 + 𝑚𝑦 𝒚 + 𝑚 𝑧 𝒛 Resolving this equation into its three separate components, we see that Fxhas to equal m 𝑥and similarly for the y and z components. That is, in Cartesian coordinates, the single vector equation (A) is equivalent to the three separate equations: In Cartesian coordinates, Newton's second law in three dimensions is equivalent to three one-dimensional versions of the same law. F= ma = m 𝒓 Fx= m 𝑥, 𝐹 𝑦 = 𝑚𝑦 and 𝐹 𝑧 = 𝑚 𝑧
• 22. Newton's Second Law in Two-Dimensional Polar Coordinates Let x and y are rectangular coordinates and r and ϕ, are polar coordinates. it is convenient to introduce two unit vectors, 𝒓 and ϕ. 𝒓 is the unit vector that points in the direction we move when r increases with ϕ fixed; likewise, ϕ is the unit vector that points in the direction we move when ϕ increases with r fixed. In figure (a) The unit vector 𝒙 points in the direction of increasing x with y fixed. (b) The unit vector 𝒓 points in the direction of increasing r with ϕ fixed; ϕ points in the direction of increasing ϕwith r fixed. Unlike 𝒙, the vectors 𝒓 and ϕ change as the position vector r moves. For anyvector r, we can define 𝒓as the unit vector in the direction of r, namely 𝒓 = r / r . Since the two unit vectors 𝒓 and ϕ are perpendicular vectors in our two-dimensional space, any vector can be expanded in terms of them. For instance, the net force F on an object can be written; …… (1)F = Fr 𝒓+ Fϕϕ r = r 𝒓To find the components of vin polar coordinates; we must differentiate with respect to t. As the vector 𝒓 changes as r moves. Thus when we differentiate it, we shall pick up a term involving the derivative of 𝒓.
• 23. In figure (a) The positions of a particle at two successive times, t1 and t2, and t2 = t1 + ∆𝑡. Unless the particle is moving exactly radially, the corresponding unit vectors 𝒓(t1 ) and 𝒓(t2 ) point in different directions. (b) The change ∆ 𝒓 in r is given by the triangle shown. If the corresponding angles ϕ(t1) and ϕ(t2) are different, then the two unit vectors 𝒓(t1) and 𝒓(t2)point in different directions. The change in 𝒓 is shown in Figure (b), and (provided ∆𝑡 is small) is approximately; ∆ 𝒓 ≈ ∆ϕϕ ≈ ϕ ∆t ϕ The direction of ∆ 𝒓 is perpendicular to 𝒓, , namely the direction of ϕ. If we divide both sides by ∆𝑡 and take the limit as ∆𝑡 → 0, then ∆ 𝒓/∆𝑡 → d 𝒓/dt and we find that; 𝑑 𝒓 𝑑𝑡 = ϕϕ ………………………………… (2) 𝑑 𝒓 𝑑𝑡 is in the direction of ϕ and is proportional to the rate of change of the angle ϕ. The derivative of r differentiate equation by using the product rule, we get, Substituting eq (2) we get, r or v, ………………… (3) r = r 𝒓 We can write the polar components of the velocity: ……………. (4) where, ω = ϕ , is angular velocity. To write down Newton's second law, we have to differentiate a second time to find the acceleration:
• 24. a = …………… (5) To complete the differentiation in (5), we must calculate the derivative of ϕ. By inspecting this figure, you should be able to convince yourself In figure (a) The unit vector ϕ at two successive times t1 and t2. (b) The change ∆ϕ, we get, ……………… (6) From eq. (5) we get; Substituting values of derivatives of unit vectors, and simplifying we get; With rconstant, both derivatives of rare zero, and we get OR where, ω = ϕ is the angular velocity and ϕ = α, is the angular acceleration. Here (rω2) is inward "centripetal" acceleration and (rα) is tangential acceleration. We can finally write down Newton’s 2nd Law in terms of polar coordinates.
• 25. It is a three-dimensional coordinate system. It specifies point position by; (i) the distance from a chosen reference axis (longitudinal axis), (ii) the direction from the axis relative to a chosen reference direction, and (iii) the distance from a chosen reference plane perpendicular to the axis. • The axis is variously called the cylindrical or longitudinal axis, • Polar axis, is the ray that lies in the reference plane, starting at the origin and pointing in the reference direction. • The distance from the axis may be called the radial distance or radius “ρ”, • The angular coordinate is sometimes referred to as the angular position or as the azimuth “𝝋". • The radius and the azimuth are together called the polar coordinates (ρ,𝝋), as they correspond to a two- dimensional polar coordinate system in the plane through the point, parallel to the reference plane. • The third coordinate may be called the height or altitude “z”, • Cylindrical coordinates are useful in connection with objects and phenomena that have some rotational symmetry about the longitudinal axis, such as water flow in a straight pipe with round cross-section, heat distribution in a metal cylinder, electromagnetic fields produced by an electric current in a long, straight wire, and so on. • They are sometimes called "cylindrical polar coordinates" and "polar cylindrical coordinates", Cylindricalcoordinatesystem
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